Ühepoolset

Ühepoolset is a term used in graph theory to describe a particular way that a subset of vertices connects to the rest of a graph. Informally, a ühepoolset is a cluster of vertices whose connections to the remainder of the graph pass through a single external vertex, called the pole.

Formally, let G = (V, E) be a graph. A nonempty proper subset S ⊂ V is a ühepoolset if the external neighbor set Γ(S) = { v ∈ V \ S | ∃ u ∈ S with {u, v} ∈ E } has cardinality exactly 1. The unique vertex in Γ(S) is called the pole of S. Internal edges within S may exist, as may edges from S to the pole; what remains crucial is that no other vertex outside S is connected to any vertex in S.

Examples can be found in simple graphs. For instance, with V = {a, b, c, d} and E = {{a, b}, {a, c}, {a, d}}, the subset S = {b, c, d} has Γ(S) = {a}, making it a ühepoolset with pole a.

Properties and applications: A ühepoolset identifies a module of a graph that attaches to the rest of the graph through a single gateway. This makes certain algorithms for decomposition, network design, and reliability analysis more straightforward. The concept is related to, but distinct from, cut-vertices and modular decomposition, offering a way to formalize “one-gateway” substructures.

Etymology and see also: The term derives from Estonian elements meaning “one” (ühe) and “pole” (pool) together with “set.” See also: graph theory, subset, cut-vertex, modular decomposition, boundary.

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